A software designer is mapping the streets for a new racing game. All of the streets are depicted as either perpendicular or parallel lines. The equation of the lane passing through $A$ and $B$ is $-7 x+3 y=-21.5$. What is the equation of the central street $P Q$?
A. $-3 x+4 y=3$
B. $3 x+7 y=63$
C. $2 x+y=20$
D. $7 x+3 y=70
Answer (2)
Rewrite the given equation − 7 x + 3 y = − 21.5 in slope-intercept form to find its slope, which is 3 7 .
Determine the slopes of the answer choices by rewriting each equation in slope-intercept form.
Identify the slope that is the negative reciprocal of 3 7 , which is − 7 3 , indicating a perpendicular relationship.
The equation with the slope − 7 3 is 3 x + 7 y = 63 , so the final answer is 3 x + 7 y = 63 .
Explanation
Problem Analysis We are given the equation of a lane as − 7 x + 3 y = − 21.5 . We need to find the equation of the central street PQ , which is either parallel or perpendicular to the given lane. We will determine the slope of the given lane and then compare it with the slopes of the answer choices to find the correct equation.
Finding the Slope of the Given Lane First, let's rewrite the given equation in slope-intercept form ( y = m x + b ) to find its slope:
− 7 x + 3 y = − 21.5
3 y = 7 x − 21.5
y = 3 7 x − 3 21.5
So, the slope of the given lane is 3 7 .
Finding Slopes of Answer Choices Now, let's find the slopes of the lines given in the answer choices by rewriting each equation in slope-intercept form:
A. − 3 x + 4 y = 3 4 y = 3 x + 3 y = 4 3 x + 4 3 Slope: 4 3
B. 3 x + 7 y = 63 7 y = − 3 x + 63 y = − 7 3 x + 9 Slope: − 7 3
C. 2 x + y = 20 y = − 2 x + 20 Slope: − 2
D. 7 x + 3 y = 70 3 y = − 7 x + 70 y = − 3 7 x + 3 70 Slope: − 3 7
Comparing Slopes and Finding the Answer If the central street PQ is parallel to the given lane, its slope must be equal to 3 7 . If PQ is perpendicular to the given lane, its slope must be the negative reciprocal of 3 7 , which is − 7 3 .
Comparing the slopes, we see that the slope of option B is − 7 3 , which is the negative reciprocal of 3 7 . Therefore, the equation of the central street PQ is 3 x + 7 y = 63 .
Final Answer The equation of the central street PQ is 3 x + 7 y = 63 .
Examples
In city planning, understanding the relationships between streets (parallel or perpendicular) is crucial for designing efficient road networks. This problem demonstrates how to determine the equation of a street that is either parallel or perpendicular to another street, given its equation. This concept is used to ensure smooth traffic flow and optimal use of space. For example, if a new street needs to be built perpendicular to an existing one, city planners use these principles to calculate the equation of the new street, ensuring it aligns correctly with the existing infrastructure. The equation of the central street PQ is 3 x + 7 y = 63 .
The equation of the central street PQ is determined by finding the slope of the given street and its relationship to the answer choices. Since the slope of the central street must be the negative reciprocal of the slope of the given lane, option B, 3 x + 7 y = 63 , is the correct answer. Hence, the answer is B: 3 x + 7 y = 63 .
;
